Cost Functions
The following cost functions are available
Manopt.costIntrICTV12
— MethodcostIntrICTV12(M, f, u, v, α, β)
Compute the intrinsic infimal convolution model, where the addition is replaced by a mid point approach and the two functions involved are costTV2
and costTV
. The model reads
Manopt.costL2TV
— MethodcostL2TV(M, f, α, x)
compute the $\ell^2$-TV functional on the PowerManifold manifold
Mfor given (fixed) data
f(on
M), a nonnegative weight
α, and evaluated at
x(on
M`), i.e.
See also
Manopt.costL2TV2
— MethodcostL2TV2(M, f, β, x)
compute the $\ell^2$-TV2 functional on the PowerManifold
manifold M
for given data f
, nonnegative parameter β
, and evaluated at x
, i.e.
See also
Manopt.costL2TVTV2
— MethodcostL2TVTV2(M, f, α, β, x)
compute the $\ell^2$-TV-TV2 functional on the PowerManifold
manifold M
for given (fixed) data f
(on M
), nonnegative weight α
, β
, and evaluated at x
(on M
), i.e.
See also
Manopt.costTV
— FunctioncostTV(M,x [,p=2,q=1])
Compute the $\operatorname{TV}^p$ functional for data x
on the PowerManifold
manifold M
, i.e. $\mathcal M = \mathcal N^n$, where $n ∈ \mathbb N^k$ denotes the dimensions of the data x
. Let $\mathcal I_i$ denote the forward neighbors, i.e. with $\mathcal G$ as all indices from $\mathbf{1} ∈ \mathbb N^k$ to $n$ we have $\mathcal I_i = \{i+e_j, j=1,\ldots,k\}\cap \mathcal G$. The formula reads
See also
Manopt.costTV
— MethodcostTV(M, x, p)
Compute the $\operatorname{TV}^p$ functional for a tuple pT
of pointss on a Manifold M
, i.e.
See also
Manopt.costTV2
— FunctioncostTV2(M,x [,p=1])
compute the $\operatorname{TV}_2^p$ functional for data x
on the PowerManifold
manifoldmanifold M
, i.e. $\mathcal M = \mathcal N^n$, where $n ∈ \mathbb N^k$ denotes the dimensions of the data x
. Let $\mathcal I_i^{\pm}$ denote the forward and backward neighbors, respectively, i.e. with $\mathcal G$ as all indices from $\mathbf{1} ∈ \mathbb N^k$ to $n$ we have $\mathcal I^\pm_i = \{i\pm e_j, j=1,\ldots,k\}\cap \mathcal I$. The formula then reads
where $c_i(\cdot,\cdot)$ denotes the mid point between its two arguments that is nearest to $x_i$.
See also
Manopt.costTV2
— MethodcostTV2(M,(x1,x2,x3) [,p=1])
Compute the $\operatorname{TV}_2^p$ functional for the 3-tuple of points (x1,x2,x3)
on the Manifold M
. Denote by
the set of mid points between $x_1$ and $x_3$. Then the functionr reads
See also
Manopt.cost_L2_acceleration_bezier
— Methodcost_L2_acceleration_bezier(M,B,pts,λ,d)
compute the value of the discrete Acceleration of the composite Bezier curve together with a data term, i.e.
where for this formula the pts
along the curve are equispaced and denoted by $t_i$ and $d_2$ refers to the second order absolute difference costTV2
(squared), the junction points are denoted by $p_i$, and to each $p_i$ corresponds one data item in the manifold points given in d
. For details on the acceleration approximation, see cost_acceleration_bezier
. Note that the Beziér-curve is given in reduces form as a point on a PowerManifold
, together with the degrees
of the segments and assuming a differentiable curve, the segmenents can internally be reconstructed.
See also
∇L2_acceleration_bezier
, cost_acceleration_bezier
, ∇acceleration_bezier
Manopt.cost_acceleration_bezier
— Methodcost_acceleration_bezier(
M::Manifold,
B::AbstractVector{P},
degrees::AbstractVector{<:Integer},
T::AbstractVector{<:AbstractFloat},
) where {P}
compute the value of the discrete Acceleration of the composite Bezier curve
where for this formula the pts
along the curve are equispaced and denoted by $t_i$, $i=1,\ldots,N$, and $d_2$ refers to the second order absolute difference costTV2
(squared). Note that the Beziér-curve is given in reduces form as a point on a PowerManifold
, together with the degrees
of the segments and assuming a differentiable curve, the segmenents can internally be reconstructed.
This acceleration discretization was introduced in[BergmannGousenbourger2018].
See also
∇acceleration_bezier
, cost_L2_acceleration_bezier
, ∇L2_acceleration_bezier
- BergmannGousenbourger2018
Bergmann, R. and Gousenbourger, P.-Y.: A variational model for data fitting on manifolds by minimizing the acceleration of a Bézier curve. Frontiers in Applied Mathematics and Statistics (2018). doi 10.3389/fams.2018.00059, arXiv: 1807.10090